Semiclassical analysis

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Master’s Thesis

Semiclassical analysis

the double well potential in the semiclassical limit

Author:

T.J. Manschot

Supervisor:

Prof. dr. N.P. Landsman

Second reader:

Dr. M.H.A.H. M¨

uger

July 28, 2020

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Abstract

Semiclassical analysis deals with the relationship between classical dynamics and the behaviour of solutions to pseudodifferential operators depending on a small parameter h > 0. Let a : R2n_{→ C}

be a function, then we can associate it with an operator aW(x, hD). In the context of quantum mechanics, taking the limit h → 0 is a way to study the classical limit of quantum mechanics. In the first part of this thesis, we will follow [8] and prove the Agmon-Lithner estimate and the Carleman inequality.

The second part of this thesis deals with double well potentials. We will study the behaviour of eigenfunctions of the Schr¨odinger operator P (h) = −h2∂2x+ V (x) where the double well potential V

is symmetric. Following Jona-Lasinio et al. [9], Helffer and Sj¨ostrand [3] and [4], and Simon [7], we will study the consequences of a small perturbation ∆V that breaks the symmetry of V .

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1 Introduction

Two topics will be discussed in this thesis. In the first part, we will discuss semiclassical analysis as presented by Zworski [8]. We will consider certain classes of functions a : R2n_{→ C, (x, ξ) 7→ a(x, ξ), and}

associate such functions with semiclassical pseudodifferential operators, i.e. pseudodifferential operators that are scaled with a small parameter h > 0. We will study the behaviour of such operators in the semiclassical limit h → 0.

In the context of quantum mechanics, we interpret R2n _{as the phase space. Then x is the position}

variable and ξ is the momentum variable, and a function a : R2n _{→ C is called a symbol. Then the}

corresponding operator aW(x, hD) is a quantum observable, and the semiclassical limit h → 0 is actually the classical limit of quantum mechanics.

An important example is the total energy function p(x, ξ) := |ξ|2+ V (x), where |ξ|2 is the kinetic energy and V is the potential. This symbol gives rise to the Schr¨odinger operator P (h) := −h2_{∆+V . We}

will prove the Agmon-Lithner estimate and the Carleman inequality for eigenfunctions of this operator in the limit h → 0.

We will need some preliminary definitions, which we will give in section 2. In section 3, we will give an overview of semiclassical Fourier analysis, which is just a rescaling of standard Fourier analysis by a parameter h > 0. The Fourier transformation will first be defined on the Schwartz space S (Rn_{). We}

will then generalise the Fourier transformation to the dual spaceS0(Rn_).

Section 4 deals with semiclassical quantisation. We will again start with symbols in S (R2n_{), which}

give rise to bounded operators L2_(Rn_{) → L}2_(Rn_). _{We will then discuss larger classes of symbols}

Sδ(m) where 0 ≤ δ ≤ 1/2 and m is a so-called order function. Such symbols give rise to

opera-tors S0(Rn) → S0(Rn). We will prove that symbols in S = S0(1) give rise to bounded operators

L2(Rn) → L2(Rn).

In section 5, we will prove several important inequalities. We will first prove the G˚arding inequality for symbols a ∈ S. Then we will prove the Agmon-Lither estimate and the Carleman estimate for eigen-functions of P (h).

In the second part, we will apply these results to a potential V that has multiple wells, following several papers from the 1980s. The main goal of this part is to provide a focused and detailed approach to the ideas presented by B. Helffer and J. Sj¨ostrand in [3] and [4] and by B. Simon in [7]. Two other important papers are Jona-Lasinio et al. [9] and Graffi et al. [10].

In section 6, we will consider the symmetric double-well potential, following Helffer [2]. The splitting of the lowest two eigenvalues of P (h) is of order ˜O(e−δ0/h_{), and the eigenfunctions corresponding to these}

eigenvalues are symmetric. In section 7, we will consider a slightly perturbed potential ˜V = V + t∆V where t ∈ [−1, 1]. Surprisingly, even a very small perturbation has drastic consequences for the eigen-functions. If ∆V is supported close to one of the wells, the eigenfunctions will be localised in just one of the wells, even if t = e−γ/h for some sufficiently small constant γ > 0. As a result, the perturbed eigenfunctions are not even approximately symmetric.

1.1 acknowledgment

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2 Introduction to the Schr¨

odinger equation

In this section, we will introduce the position and momentum operators, as well as the Schr¨odinger operator P (h) and its eigenfunctions.

2.1 Wave functions

In classical physics, a particle’s state is just its position x and its momentum p := m ˙x. These develop over time according to the differential equation F = m∂2

tx, where ∂t := _dtd. In quantum mechanics, a

particle’s position and momentum are not localised as points in phase-space. Instead, a particle’s state is given by its so-called wave function u : R × Rn→ C, (t, x) 7→ u(t, x).

This wave function u is a probability amplitude in the sense that |u(x)|2= u(x)u(x) is a probability density function, i.e. the probability of finding the particle in U ⊆ Rn _{is given by kuk}

L2_{(U )}. For this

reason, we have u ∈ L2_(Rn_{) and kuk}

L2_(Rn₎= 1. Moreover, wave functions solve the Schr¨odinger equation

i~∂tu(t, x) = −~ 2

2m∆u(t, x) + V (x)u(t, x) (2.1) where ~ is the Planck constant, m the particle’s mass, and the map V : Rn → R is the potential.

We will simplify this equation by setting m = 1/2 and replacing ~ with a dimensionless constant h > 0. Moreover, we will consider stationairy states, i.e. functions u such that ih∂tu ≡ 0.

Definition 2.1. (Schr¨odinger operator) Let h > 0 and let V : Rn → R be a smooth function not depending on h, then the Schr¨odinger operator P (h) is defined by

P (h)u = −h2∆u + V u, (2.2) In subsections 2.3 and 2.4, we will define the appropriate domain of P (h) as well as what it means for a function u to solve the time-independent Schr¨odinger equation P (h)u = E(h)u.

2.2 Position and momentum operators

We will now consider the position operator Xj and the momentum operator Pj, where 1 ≤ j ≤ n. For

any wave function u ∈ L2_(Rn_{), kuk = 1, the expectation values are hu, X}

jui and hu, Pjui. We will use

the convention that

Dxj := 1 i ∂ ∂xj . Then hu, Xjui = Eu(Xj) = Z Rn dxxj|u(x)|2 = hu, xjuiL2, hu, Pjui = Eu(Pj) = 1 2∂tEu(Xj) = Z Rn dx 1 2xj∂t(u(x)u(x)) = Z Rn dx 1 2xj

(∂tu(x))u(x) + u(x)(∂tu(x))

= Z Rn dx 1 2xj 1 −ih(−h

2_{∆u(x) + V (x)u(x))u(x) + u(x)}1

ih(−h 2_{∆u(x) + V (x)u(x))} = Z Rn dx h 2ixj ∆u(x)u(x) − u(x)∆u(x) = Z Rn dx h 2xjiD · u(x)Du(x) − u(x)Du(x) = − Z Rn dx h 2ej· u(x)Du(x) − u(x)Du(x) = Z Rn dxhu(x)hDxju(x) i = hu, hDxjuiL2.

Hence Xju = xju and Pju = hDu. In order to avoid confusion, we will always denote the momentum

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2.3 Sobolev spaces

We have seen that wave functions u are quadratically integrable, i.e. u ∈ L2_(Rn_{), and that they need to}

satisfy the Schr¨odinger equation (equation2.1). However, most functions in L2_(Rn_{) are not differentiable.}

To overcome this problem, we will recall the notion of weak derivative and the Sobolev function space Hk_(Rn_).

Definition 2.2. (Test functions) Let U ⊆ Rn _{be open, then a function ϕ : U → C is called a test function}

if it is smooth and compactly supported. The space of such functions is denoted by C_c∞(U ).

Note that test functions vanish at the boundary ∂U . This follows from the fact that they are sup-ported on a compact subset of U and the fact that U is open.

Now let ϕ ∈ C∞

c (U ) and let α := (α1, . . . , αn) be a multi-index (i.e. α ∈ Nn) such that |α| :=

α1+ . . . + αn≤ k. If we assume that u ∈ Ck(U ), then the partial derivative

Dαu := 1 i|α| ∂α1 ∂xα1 1 . . . ∂ αn ∂xαn n u (2.3)

exists, and we can obtain the equality Z U dx[u(x)Dαϕ(x)] = (−1)|α| Z U dx[Dαu(x)ϕ(x)]

by integrating by parts |α| times. In case u /∈ Ck_{(U ), we will use this equation to define generalise D}α_u.

Let v be some function. We want Z U dx[u(x)Dαϕ(x)] = (−1)|α| Z U dx[v(x)ϕ(x)],

because then we can set Dαu := v. Clearly, these integrals can only exist if the functions u and v are integrable on Supp(ϕ) for all ϕ ∈ Cc∞(U ). This motives the following two definitions.

Definition 2.3. (Locally integrable functions) Let U ⊆ Rn _{be open and let u : U → C be a function.}

Then u is called locally integrable if

u|V ∈ L1(V )

for all open V ⊂⊂ U , i.e. for all open V ⊆ U such that V is compact and V ⊂ U . The set of locally integrable functions is denoted L1

loc(U ).

Definition 2.4. (Weak derivatives) Let u, v ∈ L1

loc(U ) and let α be a multi-index, then Dαu := v is

called the weak αth _{partial derivative of u if}

Z U dx[u(x)Dαϕ(x)] = (−1)|α| Z U dx[v(x)ϕ(x)] for all ϕ ∈ C_c∞(U ).

Lemma 2.5. (Uniqueness of weak derivatives) Weak derivatives are unique up to sets of Lebesgue-measure zero.

Proof. Let u, v, v0∈ L1

loc(U ) and let α := (α1, . . . , αn) be a multi-index. Assume that v and v0 are weak

αth partial derivatives of u. Then we have for all test functions ϕ ∈ Cc∞(U ) that

(−1)|α| Z U dx[v(x)ϕ(x)] = Z U dx[u(x)Dαϕ(x)] = (−1)|α| Z U dx[v0(x)ϕ(x)].

But then we have

Z

U

dx[(v(x) − v0(x))ϕ(x)] = 0 for all ϕ ∈ C_c∞(U ), hence v = v0 almost everywhere.

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Of course, weak derivatives need not exist in general. We will consider spaces of square integrable functions that have weak derivatives up to some degree k ∈ N.

Definition 2.6. (Sobolev spaces) Let U ∈ Rn _{be open and let k ∈ N, then the Sobolev space H}k_{(U )}

consists of all functions u ∈ L2_{(U ) have weak derivatives D}α_{u ∈ L}2_{(U ) for all multi-indices α such that}

|α| ≤ k.

We can define an inner product on Hk_{(U ) by}

hu, viHk_{(U )}:=

X

|α|≤k

hDα_{u, D}α_vi

L2_{(U )}, (2.4)

making Hk(U ) into a Hilbert space.

2.4 Weak solutions

The definition of weak solutions u of P (h)u = E(h)u is analogous to the definition of the weak derivative. Let u such that P (h)u = E(h)u and let v be some function. Then

0 = hv, (P (h) − E(h))ui =

Z

Rn

dxhv(x) −h2∆u(x) + V (x)u(x) − E(h)u(x)i

= −h2 n X j=1 Z Rn dxhv(x)∂_j2u(x)i+ Z Rn dxhv(x)(V (x) − E(h))u(x)i = h2 n X j=1 Z Rn dxh∂jv(x)∂ju(x) i + Z Rn dxhv(x)(V (x) − E(h))u(x)i = hv, (V − E(h))ui + n X j=1 hhDjv, hDjui.

Definition 2.7. (Weak solutions of the Schr¨odinger equation) A function u ∈ H1_(Rn_{) possibly depending}

on h is called a weak solution of P (h)u = E(h)u if

hv, (V − E)ui +

n

X

j=1

hhDjv, hDjui = 0 (2.5)

for all v ∈ C_c∞(Rn_{). A function u ∈ H}1_(Rn_{) is called a solution to the Schr¨}_{odinger equation if it is a}

weak solution of P (h)u = E(h)u and kukL2_(Rn₎= 1.

Since we mostly deal with the momentum operator hD instead of the differential operator D, it makes sense to scale the Sobolev norm accordingly.

Definition 2.8. (Semiclassical Sobolev norm) Let U ⊆ Rn be open. The semiclassical Sobolev norm on the space Hk(U ) is given by

kukHk h(U ):=   X |α|≤k khDαuk2_L2_{(U )}   1/2 . (2.6)

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3 Semiclassical Fourier analysis

In this section, we will discuss semiclassical Fourier theory. The semiclassical Fourier transformFh is a

rescaling of the standard Fourier transform using the parameter h > 0. ThenFh maps ϕ(x) into ˆϕh(ξ)

such that (hD)α_{ϕ 7→ ξ}α_ϕ_ˆ

h and (−x)αϕ 7→ (hD)αϕˆh.

3.1 Schwartz space

S (R

n

₎

Our goal is to define the semiclassical Fourier transform on a large class of functions, but we will start with just the so-called Schwartz functions. Schwartz functions behave very nicely in the sense that all their derivatives decrease rapidly as |x| → ∞, as does the function itself. An example of such a function is the Gaussian function ϕ(x) := e−π|x|2

.

Definition 3.1. (Schwartz space) The Schwartz spaceS (Rn_{) consists of all functions ϕ ∈ C}∞_(Rn_{) such}

that

sup

x∈Rn

|xα_∂β_{ϕ(x)| < ∞}

for all multi-indices α, β.

Note that kϕkα,β := supx∈Rn|xα∂βϕ(x)| is a seminorm onS (Rn) for each pair of multi-indices α, β.

This collection of seminorms defines a topology onS as follows. Consider V (α, β, k) := {ϕ ∈S (Rn) | kϕkα,β <

1 k}.

The collection of finite intersections of such sets is a convex, balanced local base of a topology inS (Rn) turning it into a locally convex space such that all seminorms are continuous.

A subset U ⊂S (Rn_{) is bounded if and only if {kϕk}

α,β | ϕ ∈ U } is bounded for all multi-indices α, β.

It should be noted that all k · kα,β are actually norms, and that the spacesS (Rn) with this norm is a

Fr´echet space.

It is easy to see that C_c∞(Rn_{) ⊂}_{S (R}n_{) ⊂ L}2_(Rn_{). Recall that C}∞

c (Rn) is dense in L2(Rn), then it

follows trivially thatS (Rn_{) is dense in L}2_(Rn_{) in the L}2_-norm.

Definition 3.2. (Semiclassical Fourier transform) Let ϕ ∈S , then its semiclassical Fourier transform ˆ ϕh is defined by ˆ ϕh(ξ) :=Fh(ϕ)(ξ) := Z Rn dxhϕ(x)e−hihx,ξi i (3.1) where ξ ∈ Rn_.

Remark 3.3. In case h = 1, we also write F := F1 and ˆϕ := ˆϕ1, which is the standard Fourier

transform. Note that some authors define the Fourier transform with a normalisation factor 1/(2πh)n/2_.

Proposition 3.4. (Properties of the semiclassical Fourier transform) The semiclassical Fourier trans-formFh:S → S is an isomorphism, whose inverse is given by

ϕ(x) = 1 (2πh)n Z Rn dξhϕˆh(ξ)e i hhx,ξi i

for all x ∈ Rn_{. Furthermore, we have the following equalities:}

(i) Fh((hDx)αϕ) = ξαFh(ϕ),

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Proof. We will only show that equalities (i) and (ii) hold. The other statements are well-known facts from standard Fourier theory. Let ξ ∈ Rn, then we have

Fh((hDx)αϕ)(ξ) = Z Rn dxh(hDx)αϕ(x)e− i hhx,ξi i = (−1)|α| Z Rn dxhϕ(x)(hDx)αe− i hhx,ξi i = (−1)|α| Z Rn dx ϕ(x) h i α −i hξ α e−hihx,ξi = ξα Z Rn dxhϕ(x)e−hihx,ξi i = ξαFh(ϕ)(ξ), (hDξ)α(Fh(ϕ))(ξ) = (hDξ)α Z Rn dxhϕ(x)e−hihx,ξi i = Z Rn dxhϕ(x)(hDξ)αe− i hhx,ξi i = Z Rn dxhϕ(x)(−x)αe−hihx,ξi i =Fh((−x)αϕ)(ξ).

Lemma 3.5. (More properties) Let ϕ, ψ ∈S , then we have Z Rn dx [Fh(ϕ)(x)ψ(x)] = Z Rn dx [ϕ(x)Fh(ψ)(x)] (3.2) and Z Rn dxhϕ(x)ψ(x)i= 1 (2πh)n Z Rn dxhFh(ϕ)(x)Fh(ψ)(x) i . (3.3) Proof. Z Rn dx [Fh(ϕ)(x)ψ(x)] = Z Rn dx Z Rn dyhϕ(y)e−hihy,xi i ψ(x) = Z Rn dy Z Rn dxhψ(x)e−hihx,yi i ϕ(y) = Z Rn dx [ϕ(x)Fh(ψ)(x)]

This proves equation (3.2). Now we can substitute ϕ withFh(ϕ) to obtain

Z Rn dxhFh(ϕ)(x)Fh(ψ)(x) i = Z Rn dxhFh(Fh(ϕ))(x)ψ(x) i . Note thatFh(ϕ)(ξ) = R Rndy h ϕ(x)ehihy,ξi i = (2πh)n_F−1 h (ϕ)(ξ), so Fh(Fh(ϕ))(x) = (2πh)nϕ(x).

Hence k ˆϕhk = (2πh)n/2kϕk. We will now prove a few norm estimates that will prove useful later.

The notation hxi := (1 + |x|2)1/2 will be convenient. Note that R_Rndx[hxi−(n+1)] < ∞ and there is a

constant C > 0 such that hxik_{≤ C max}

|α|≤k|xα| for all x ∈ Rn, k ∈ N.

Lemma 3.6. (norm estimates) Let u ∈S and h > 0, then there is some constant C > 0 such that kûhkL∞ ≤ kuk_L1 (3.4) kukL∞ ≤ 1 (2πh)nkûhkL1 (3.5) kûhkL1 ≤ C max |α|≤n+1k∂ α_uk L1 (3.6)

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3.2 Tempered distributions

S

0

(R

n

)

The Schwartz space is very small class of functions, so our goal is to extend the Fourier transform to a wider class of functions. Let u : Rn _{→ C be a quadratically integrable function. Then we can define a}

continuous linear map u : S → C by u(ϕ) := R_Rndx [u(x)ϕ(x)]. This will serve as motivation for the

following definition.

Definition 3.7. (Tempered distributions) The set

S0_(Rn_{) := {u :}_{S → C | u is linear and continuous }} _(3.7)

is the dual space ofS (Rn_{) and maps u ∈}_S0 _{are called tempered distributions.}

A sequence {uj}j∈N⊂S0 is said to converge inS0 if there is a map u ∈S0 such that {uj(ϕ)}j∈N⊂ C

converges to u(ϕ) for all ϕ ∈S . For u ∈S0, we will sometimes write

Z

Rn

dx [u(x)ϕ(x)] := u(ϕ)

even though such a locally integrable function u : Rn→ C does not exist in general.

Definition 3.8. (More on tempered distributions) Let u ∈S0 and ϕ ∈S . Let α be a multi-index and let x ∈ Rn, then we define:

(i) Dαu(ϕ) := (−1)|α|u(Dαϕ), in the same spirit as the weak derivative. Note that Dαϕ and hence Dα_{u is guaranteed to exist.}

(ii) (xα_{u)(ϕ) := u(x}α_ϕ)

(iii) We take equation (3.2) as a definition for the semiclassical Fourier transform on tempered distri-butions, i.e. (Fhu)(ϕ) := u(Fhϕ).

Example 3.9. (Dirac delta function) Let x0∈ Rn, then we define the Dirac delta function δx0 at x0 by

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at x0 and zero everywhere else, such that its integral is

R

Rndx[δx0(x)] = 1. Then we have for any ϕ ∈S

that R_Rndx[δx0(x)ϕ(x)] = ϕ(x0).

We can calculate its Fourier transform by

(Fhδx0)(ϕ) := δx0(Fhϕ) =Fhϕ(x0) = Z Rn dx[ϕ(x)e−hihx,x0i_]. In other words,Fhδx0(x) := e −i hhx,x0i_{. In particular,}F h(δ0) ≡ 1.

Many of the properties ofFh:S (Rn) →S (Rn) hold more generally.

Remark 3.10. (Semiclassical Fourier transform on L2_{) Let u ∈ L}2_(Rn_{), then we can interpret u as a}

tempered distribution defined by

ϕ 7→ Z

Rn

dx [u(x)ϕ(x)]

where ϕ ∈S . Then we have for the semiclassical Fourier transform ˆuh of u ∈ L2 that

Z Rn dξ [ˆuh(ξ)ϕ(ξ)] := Z Rn dx [u(x) ˆϕh(x)] = Z Rn dx Z Rn dξhu(x)ϕ(ξ)e−hihx,ξi i = Z Rn dξ Z Rn dxu(x)e−hihx,ξi ϕ(ξ)

for all ϕ ∈ S . So equation (3.1) is still valid for L2_{-functions. The same is true for the inverse}

semiclassical Fourier transform. As a result, lemma3.5holds for L2_{-functions as well.}

Proposition 3.11. (Properties of the semiclassical Fourier transform on tempered distributions) Let u ∈S0, x, ξ ∈ Rn, and let α be a multi-index. Then we have

(i) Fh((hDx)αu) = ξαFh(u), and

(ii) Fh((−x)αu) = (hDξ)αFh(u).

Proof. For all ϕ ∈S , we have

Fh((hD)αu) = (hD)αu(Fh(ϕ)) = (−1)|α|u((hD)αFh(ϕ)) = (−1)|α|u(Fh((−x)αϕ))) =Fh(u)(ξαϕ) = ξαFh(u)(ϕ) Fh((−x)αu)(ϕ) = (−ξ)αu(Fh(ϕ)) = u((−ξ)αFh(ϕ)) = (−1)αu(Fh((hD)αϕ)) = (−1)αFh(u)((hD)αϕ) = (hD)αFh(u)(ϕ)

3.3 Uncertainty principle

In this subsection, we will prove Heisenberg’s uncertainty principle. Let u ∈S (Rn) such that kuk = 1 and let 1 ≤ j ≤ N . Consider the standard deviation of position σxj and the standard deviation of

momentum σhDj. Note that

σ2_x_j = hu, x2_jui − hu, xjui2,

σ2hDj = hu, (hDj)

2_{ui − hu, hD} jui2.

Then Heisenberg’s uncertainty principle states that σxjσhDj ≥ h/2. We interpret that the position and

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The relation between a wave function u(x) and its semiclassical Fourier transform ˆuh(ξ) is

charac-terised by equation (3.3), i.e. for u ∈S (Rn), the expectation values for the position and momentum operators in direction 1 ≤ j ≤ n are given by

hu, xjui = Z Rn dxhu(x)xju(x) i = 1 (2πh)n Z Rn dξhFh(u)(ξ)Fh(xju)(ξ) i = 1 (2πh)n Z Rn dξhuˆh(ξ)(−hDj)ûh(ξ) i = 1 (2πh)nhûh, −hDjuˆhi, hu, hDjui = Z Rn dxhu(x)hDju(x) i = 1 (2πh)n Z Rn dξhFh(u)(ξ)Fh(hDju)(ξ) i = 1 (2πh)n Z Rn dξhuˆh(ξ)ξjuˆh(ξ) i = 1 (2πh)nhûh, ξjuˆhi.

For this reason, u(x) is said to be the wave function in position coordinates, and ˆuh(ξ) the wave function

in momentum coordinates.

Lemma 3.12. (Shifting the position coordinates) Let u ∈S (Rn_{) such that kuk}

L2_(Rn₎= 1 and let a ∈ Rn,

then we can shift the coordinates by setting x y := x − a and v(y) := u(y + a). The shifted wave function v satisfies hv, yvi = hu, xui − a and hv, hDvi = hu, hDui.

Proof. This is a fairly straightforward computation. Let 1 ≤ j ≤ n, then hv, yjvi = Z Rn dyhv(y)yjv(y) i = Z Rn

dyhu(y + a)yju(y + a)

i = Z Rn dxhu(x)(xj− aj)u(x) i = hu, xjui − aj, ˆ vh(ξ) = Z Rn dyhv(y)e−hihy,ξi i = Z Rn

dyhu(y + a)e−hihy,ξi

i = Z Rn dxhu(x)e−hihx−a,ξi i = ehiha,ξi_u_ˆ_h_(ξ), hˆvh, ξjˆvhi = Z Rn dξhvˆh(ξ)ξjˆvh(ξ) i = Z Rn dξhehiha,ξiuˆ_h(ξ)ξ_je i hha,ξiuˆ_h(ξ) i = Z Rn dξhuˆh(ξ)ξjuˆh(ξ) i = hˆuh, ξjuˆhi.

Lemma 3.13. (Shifting the momentum coordinates) Let u ∈S (Rn_{) such that kuk}

L2_(Rn₎and let b ∈ Rn,

then we can shift the coordinates by setting ξ η := ξ − b and ˆvh(η) := ˆuh(η + b). The shifted wave

function v satisfies hv, xvi = hu, xui and hv, hDvi = hu, hDui − b. Proof. This is analogous to the previous lemma.

Corollary 3.14. By setting a := hu, xui and b := hu, hDui, we obtain a shifted wave function v such that hv, xvi = hv, hDvi = 0. So for any u ∈S (Rn) such that kukL2_(Rn₎, we can assume without loss of

generality that σ2_x_j = hu, x2_jui = kxjuk2, σ2_hD j = hu, (hDj) 2_{ui = khD} juk2.

From now on, we will write kxjuk and khDjuk = (2πh)−n/2kξjuˆhk instead of σxj and σhDj. Only

one more lemma is needed before the uncertainty principle can be proved.

Lemma 3.15. (Commutation relation of position and momentum) The commutation relation of the position and momentum operators xj and hDj as operatorsS (Rn) →S (Rn) is given by

[xj, hDj] := xjhDj− hDjxj = ih.

Proof. Let u ∈S (Rn_{). Then we can simply compute that}

[xj, hDj]u = xjhDju − hDj(xju) =

h

i(xj∂xju − ∂xj(xju)) = −

h

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Proposition 3.16. (Heisenberg uncertainty principle) Let u ∈S (Rn), then kxjukkξjuˆhk ≥

h

2kukkˆuhk. (3.8) In particular, if kukL2_(Rn₎= 1, then

kxjukkhDjuk ≥

h

2. (3.9)

Proof. Let u ∈S (Rn_{). Using the Cauchy-Schwarz inequality and the previous lemma we obtain}

kxjukkhDjuk ≥ |hhDju, xjui| ≥ |=hhDju, xjui|

= 1 2|hhDju, xjui − hxju, hDjui| = 1 2|h(xjhDj− hDjx)u, ui| = 1 2|h[xj, hDj]u, ui| = h 2kuk 2_, hence kxhukkξjuˆhk = kxjuk(2πh)n/2khDjuk ≥ h 2(2πh) n/2_kuk2₌h 2kukkˆuhk.

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4 Semiclassical quantisation

In classical mechanics, the state of a system is completely determined by the position and momentum variables, and all dynamical quantities are a function of said variables. In the context of semiclassical quantisation, such a function is called a symbol. Examples are: the kinetic energy T := ξ2_{or the angular}

momentum L := x × ξ (where × is the outer product on R3_).

In quantum mechanics, the operators associated with the position and momentum are x and hD, respectively. This raises the question what operators are associated with the other symbols. In this section, we will discuss this for various classes of symbols.

Definition 4.1. (Symbols) A function a : R2n→ R, (x, ξ) 7→ a(x, ξ) is called a symbol.

Since x and hD do not commute, we can immediately see that there is no canonical way to quantise symbols that are linear in both arguments, such as a(x, ξ) := hx, ξi. We could pick any linear combination

a(x, ξ) = thx, ξi + (1 − t)hξ, xi. This motivates the following definition.

Definition 4.2. (Semiclassical quantisation) Let t ∈ [0, 1] and let a(x, ξ) be a symbol, then the semi-classical pseudodifferential operator Opt(a) is defined by

Opt(a)u(x) := 1 (2πh)n Z Rn dξ Z Rn

dyha(tx + (1 − t)y, ξ)u(y)ehihx−y,ξi

i

. (4.1)

In particular, we will be interested in the standard quantisation a(x, hD) := Op1(a) and the Weyl

quan-tisation aW_{(x, hD) := Op} 1 2(a).

Remark 4.3. In subsections 4.1, we will show that Opt(a) : L2(Rn) → L2(Rn) is indeed well-defined for

symbols a ∈S (R2n_{). In subsection 4.3, we will show that Op}

t(a) :S0(Rn) →S0(Rn) is well-defind for

symbols a ∈ Sδ(m).

The Weyl quantisation is the most important quantisation formula because it gives rise to a self-adjoint operator if the symbol a is real-valued. The standard quantisation is important because

a(x, hD)u(x) = 1 (2πh)n Z Rn dξ Z Rn

dyha(x, ξ)u(y)ehihx−y,ξi

i = 1 (2πh)n Z Rn dξha(x, ξ)Fhu(ξ)e i hhx,ξi i =F_h−1(a(x, ·)Fhu(·))(x),

for a ∈S (R2n) and u ∈S (Rn). This makes that standard quantisation easier to compute. The other Opt are useful because they allow us to transfer computations from the standard quantisation to the

Weyl quantisation, as we will see in the proof of lemma4.38.

4.1 Semiclassical quantisation for a ∈

S (R

2n

₎

The following defintion gives a very convenient way to abbreviate the formula for Opt(a).

Definition 4.4. (Kernel of Opt) Let t ∈ [0, 1], then we define the kernel Kt of Opt by

Kt(x, y) :=

1 (2πh)n

Z

Rn

dξha(tx + (1 − t)y, ξ)ehihx−y,ξi

i =F_h−1(a(tx + (1 − t)y, ·))(x − y).

Note that Opt(a)u(x) =R_Rndy [Kt(x, y)u(y)].

Now assume that a ∈ S (R2n). SinceF_h−1 :S (Rn) → S (Rn), we also have Kt(x, ·) ∈S (Rn). So

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Lemma 4.5. Let a ∈S (R2n) be a symbol, t ∈ [0, 1], then Opt(a) :S0→S is continuous.

Proof. Consider a sequence uj → u converging inS0(Rn) and let α, β be multi-indices. Then

xα∂β(u(Kt(x, ·)) − uj(Kt(x, ·))) = u(xα∂βKt(x, ·)) − uj(xα∂βKt(x, ·)) → 0,

hence Opt(a)uj → Opt(a)u inS (Rn) and Opt(a) is indeed continuous.

Proposition 4.6. Let a ∈S (R2n_{) and let h > 0, then the operator}

aW(x, hD) : L2(Rn) → L2(Rn)

is bounded uniformly in h, i.e. there is some constant C > 0 not depending on h such that we have kaW_{(x, hD)uk ≤ Ckuk for all u ∈ L}2_(Rn_).

Proof. We have a ∈S (R2n_{), and so K}

1/2∈S (R2n). Define the two constants

C1:= sup x∈Rn Z Rn dy|K1/2(x, y)| < ∞, C2:= sup y∈Rn Z Rn dx|K1/2(x, y)| < ∞,

then the L2_{-norm of a}W_{(x, hD)u is}

kaW_{(x, hD)uk}2_{= ha}W_{(x, hD)u, a}W_{(x, hD)ui}

≤ Z Rn dx Z Rn dy Z Rn dz|K1/2(x, y)||u(y)||K1/2(x, z)||u(z)| ≤ Z Rn dx Z Rn dy Z Rn dz |K1/2(x, y)||K1/2(x, z)| 1 2 |u(y)| 2_{+ |u(z)|}2 = Z Rn dx Z Rn dy Z Rn dz|K1/2(x, y)||K1/2(x, z)||u(y)|2 ≤ C1C2 Z Rn dy|u(y)|2 = C1C2kuk2

Theorem 4.7. Let a ∈S (R2n), then the operator aW(x, hD) : L2→ L2 _{is compact.}

Proof. Recall the definition of a compact operator (see C.1). The operator aW_{(x, hD) is compact if}

for any bounded sequence {uk}k∈N ⊂ L2(Rn), the sequence {aW(x, hD)uk}k∈N ⊂ L2(R2n) has some

converging subsequence.

Let {uk}k∈N⊂ L2(Rn) be a bounded sequence and let k, l ∈ N. We want to find a subsequence {u0k}

of {uk} such that the sequence {aW(x, hD)u0k} converges. Let N ∈ N be some fixed constant (we will

later choose N > n/2), then we have for some sufficiently large constant C > 0 that kaW_{(x, hD)u} k− aW(x, hD)ulkL2 = Z Rn dx|aW_{(x, hD)u} k(x) − aW(x, hD)ul|2 = Z Rn

dxhxi−2N_|hxiN_(aW_{(x, hD)u}

k(x) − aW(x, hD)ul(x))|2

≤ CkhxiN_(aW_{(x, hD)u}

k− aW(x, hD)ul)kL∞.

So it suffices to show that the sequence {hxiN_aW_{(x, hD)u}0

k} converges in the sup-norm. We will first

construct a candidate subsequence {u0_k} and then prove that hxiN_aW_{(x, hD)u}0

k indeed converges

uni-formly.

For any x ∈ Rn, the sequence {hxiNaW(x, hD)uk(x)}k∈N ⊂ C is bounded and therefore admits a

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s.t. Qn ₌ S

p∈N{qp}. We inductively define subsequences {u (j)

k }k∈N, j ∈ N, such that {u (j)

k }k∈N ⊂

{u(j−1)_k }k∈N such that {hqjiNaW(x, hD)u (j)

k (qj)}k∈N converges. Define {u0k}k∈N by u0k := u (k) k , then

{hqiN_aW_{(x, hD)u}0

k(q)}k∈Nconverges for all q ∈ Qn.

Let > 0. Our goal is to show that there is some K ∈ N such that for all k, l ≥ K and all x ∈ Rn, |hxiN_aW_{(x, hD)u}0

k(x) − hxi

N_aW_{(x, hD)u}0

l(x)| < . Choose N > n/2, and let α, β be multi-indices. Due

to a ∈S (R2n) there exists some C > 0 s.t. sup x∈Rn |xα_∂β_(aW_{(x, hD)u)| ≤} _sup (x,y)∈R2n |xα_∂β xhyi N_{K(x, y)|} Z Rn dyhyi−N_|u(y)| ≤ CkukL2,

where we used the Cauchy-Schwartz inequality. The sequence {u0_k}k∈N is bounded in the L2-norm, so

there is some M > 0 such that for all k ∈ N, x ∈ Rn, |∂hxiN_aW_{(x, hD)u}0

k(x)| < M/3, hxiN +1|aW(x, hD)u0k(x)| < M/2.

We will consider two cases: where x is inside some open neighbourhood of 0, and where x is far away from 0. Let R > 0 be large enough such that M/R ≤ . Then

sup |x|≥R |hxiN_aW_{(x, hD)u}0 k(x) − hxi N_aW_{(x, hD)u}0 l(x)| ≤ R−1 sup |x|≥R |hxiN +1_aW_{(x, hD)u}0 k(x)| + R −1 _sup |x|≥R |hxiN +1_aW_{(x, hD)u}0 l(x)| < M/R < .

Finally, {B(q, /M )}q∈Qnis an open cover of B(0, R), and so there is a finite subcover {B(q_p, /M )}_1≤p≤P.

For all 1 ≤ p ≤ P , {hqpiNaW(x, hD)u0k(qp)}k∈N converges. So there is some K ∈ N such that for all

k, l ≥ K, 1 ≤ p ≤ P ,

|hqpiNaW(x, hD)u0k(qp) − hqpiNaW(x, hD)u0l(qp)| < /3.

For any x ∈ B(0, R), choose 1 ≤ p ≤ P such that x − qp< /M , then for all k, l ≥ K;

|hxiN_aW_{(x, hD)u}0 k(x) − hxi N_aW_{(x, hD)u}0 l(x)| ≤ |hxiN_aW_{(x, hD)u}0 k(x) − hqpiNaW(x, hD)u0k(qp)| + |hqpiNaW(x, hD)u0k(qp) − hqpiNaW(x, hD)u0l(qp)

+ |hxiNaW(x, hD)u0_l(x) − hqpiNaW(x, hD)u0l(qp)|

< /3 + |x − qp|(sup x |∂hxiN_aW_{(x, hD)u}0 k(x)| + sup x |∂hxiN_aW_{(x, hD)u}0 l(x)|) < .

Proposition 4.8. (Formal adjoint) Let t ∈ [0, 1] and let a ∈S (R2n). Then the formal adjoint of Opt(a)

is Op1−t(a), i.e. for all u, v ∈ L2(Rn),

hu, Opt(a)vi = Z Rn dxhu(x)Opt(a)v(x) i = Z Rn dxhOp1−t(a)u(x)v(x) i = hOp1−t(a)u, vi.

Proof. Let Kt be the kernel of Opt(a), then its complex conjugate is

Kt(x, y) =

1 (2πh)n

Z

Rn

dξha(tx + (1 − t)y, ξ)e−hihx−y,ξi

i

= 1 (2πh)n

Z

Rn

dξha((1 − t)y + (1 − (1 − t))x, ξ)ehihy−x,ξi

i ,

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which is precisely the kernel of Op1−t(a). Hence hu, Opt(a)vi = Z Rn dxhu(x)Opt(a)v(x) i = Z Rn dx Z Rn dyhu(x)Kt(x, y)v(y) i = Z Rn dx Z Rn dyhKt(x, y)u(x)v(y) i = Z Rn dyhOp1−t(a)u(y)v(y) i = hOp1−t(a)u, vi.

Lemma 4.9. Let a ∈S0(Rn_{× R}n_{), then Op}

t(a) :S → S0.

Proof. If a ∈S0(Rn_×Rn_{), then K}

t∈S0(Rn×Rn). Now let u, v ∈S (Rn) and define v ⊗u ∈S (Rn×Rn)

by v ⊗ u(x, y) := v(x)u(y). Then we can define Opt(a)u ∈S0(Rn) by Opt(a)u(v) := Kt(v ⊗ u).

4.2 Composition of the Weyl quantisation

Let a, b be symbols, then we can ask what should be the symbol c such that cW_{(x, hD) = a}W_{(x, hD)b}W_{(x, hD).}

We will denote this symbol by a#b := c. In this subsection, we will prove that a#b(z) = e2hiσ(hDz,hDw)_(a(z)b(w))|

z=w.

In order to prove this, we will decompose symbols into Fourier components. The following lemmas will be useful.

Definition 4.10. (Linear symbols) A symbol l of the form l(z) := hz∗_{, zi = hx}∗_{, xi + hξ}∗_{, ξi for some}

z∗= (x∗, ξ∗) ∈ R2n _{is called a linear symbol. We will identify linear symbols l with their point z}∗_{∈ R}2n_.

Now let a ∈S (R2n_{), then its semiclassical Fourier transform and its inverse are}

ˆ ah(l) = Z R2n dzha(z)e−hil(z) i , a(z) = 1 (2πh)2n Z R2n dlhˆah(l)e i hl(z) i ,

and so the quantisation is

Opt(a) = 1 (2πh)2n Z R2n dlhˆah(l)Opt ehil(·) i .

for all t ∈ [0, 1]. The following lemmas deal with the quantisation and composition of such exponentials. Lemma 4.11. (Quantisation of linear symbols) Consider the linear symbol l(x, ξ) := hx∗, xi + hξ∗, ξi, then we have for all t ∈ [0, 1] that

Opt(l)u(x) = (hx∗, xi + hξ∗, hDi)u(x) (4.2)

Proof. This is just a simple calculation. Opt(l)u(x) = 1 (2πh)n Z Rn dξ Z Rn

dyhehihx−y,ξil(tx + (1 − t)y, ξ)u(y)

i = thx∗, xiu(x) + (1 − t)F_h−1◦Fh(hx∗, ·iu(·))(x) +Fh−1(hξ

∗_{, ·iˆ}_u h(·))(x)

= hx∗, xiu(x) + hξ∗, hDxiu(x).

Lemma 4.12. (Quantisation of exponentials of linear symbols) Let l be a linear symbol, and define the symbol a(z) := ehil(z). Then we have

Opt(a)u(x) := e i hhx ∗_,xi+i h(1−t)hx ∗_,ξ∗_i u(x + ξ∗). (4.3) for all t ∈ [0, 1].

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Proof. Again, we can simply calculate Opt(a)u(x) = 1 (2πh)n Z Rn dξ Z Rn

dyhehihx−y,ξi_ehil(tx+(1−t)y,ξ)_u(y)

i = 1 (2πh)n Z Rn dξ Z Rn dyhehihx−y,ξie i hhx ∗_{,tx+(1−t)yi+}i hhξ ∗_,ξi u(y)i = ehithx ∗_,xi 1 (2πh)n Z Rn dξ Z Rn dyhehihx+ξ ∗_−y,ξi ehi(1−t)hx ∗_,yi u(y)i = ehithx ∗_,xi ehi(1−t)hx ∗_,x+ξ∗_i u(x + ξ∗) = ehihx ∗_,xi+i h(1−t)hx ∗_,ξ∗_i u(x + ξ∗). Remark 4.13. We can write

ehil(x,hD)u(x) = e i h(hx ∗_,xi+1 2hx ∗_,ξ∗_i) u(x + ξ∗), so that ehil(·) W (x, hD) = ehil(x,hD) _(4.4)

Proof. Let u ∈ S (Rn_{) and t ∈ R. The partial differential equation} h

i∂tv(x, t) = l(x, hD)v(x, t) with

boundary condition v(x, 0) = u(x) has a unique solution, but it is solved by v(x, t) = eithl(x,hD)u(x) as well as by v(x, t) = ehi(thx ∗_,xi+t2 2hx ∗_,ξ∗_i) u(x + tξ∗), hence these expressions must coincide.

We will now find a#b for exponentials of linear symbols. Then we can generalise this to arbitrary a, b ∈S (R2n) by using the Fourier decomposition of a and b.

Lemma 4.14. (Composition of exponentials of linear symbols) Let l, m ∈ S (R2n) be linear, i.e. l = (x∗1, ξ1∗) and m = (x∗2, ξ∗2) for some (x∗1, ξ∗1), (x∗2, ξ2∗) ∈ R2n, then

ehil(x,hD)e i hm(x,hD)= e i 2hσ(l,m)e i h(l+m)(x,hD), (4.5) where σ(l, m) := hx∗₂, ξ₁∗i − hx∗

1, ξ2∗i is the standard symplectic product on R2n.

Proof. ehil(x,hD)e i hm(x,hD)u(x) = e i hl(x,hD)e i hhx ∗ 2,x+12ξ ∗ 2i_{u(x + ξ}∗ 2) = ehihx ∗ 1,x+12ξ ∗ 1i_ehihx ∗ 2,x+ξ ∗ 1+12ξ ∗ 2i_{u(x + ξ}∗ 2+ ξ1∗) = e2hi(hx ∗ 2,ξ ∗ 1i−hx ∗ 1,ξ ∗ 2i)_ehihx ∗ 1+x ∗ 2,x+12ξ ∗ 1+12ξ ∗ 2i_{u(x + ξ}∗ 1+ ξ ∗ 2)b = e2hiσ(l,m)_ehi(l+m)(x,hD)_.

Theorem 4.15. (Fourier decomposition of aW) Let a ∈S (R2n) and l ∈ R2n, then aW(x, hD) = 1 (2πh)2n Z R2n dlhˆah(l)e i hl(x,hD) i . (4.6)

Moreover, if a ∈S0(R2n_{) and u, v ∈}_{S (R}n_{), then we can view e}i

hl(x,hD)u as a tempered distribution by setting ehil(x,hD)_{u(v) :=} Z Rn dxhehil(x,hD)_u(x)v(x) i , which is itself in S (R2n) as a function of l, so

aW(x, hD)(u)(v) = 1 (2πh)2nˆah l 7→ ehil(x,hD)u(v . (4.7)

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Theorem 4.16. (Composition) Let a, b ∈S (R2n), then the symbol a#b defined by a#b(z) := e2hiσ(hDz,hDw)_(a(z)b(w))|

z=w (4.8)

satisfies

(a#b)W(x, hD) = aW(x, hD)bW(x, hD). (4.9) Proof. Recall that σ(hDz, hDw) = hhDξ, hDyi − hhDη, hDxi, so

e2hiσ(hDz,hDw)_ehi(l(z)+m(w)) = ∞ X k=0 " 1 k! _i 2h k σ(hDz, hDw)ke i h(l(z)+m(w)) # = ∞ X k=0 " 1 k! _i 2h k (hhDξ, hDyi − hhDη, hDxi)ke i h(l(z)+m(w)) # = ∞ X k=0 " 1 k! _i 2h k ehi(l(z)+m(w))_(hξ∗ 1, x∗2i − hξ∗2, x∗1i) k # = ∞ X k=0 " 1 k! _i 2h k σ(l, m)kehi(l(z)+m(w)) # = e2hiσ(l,m)_ehi(l(z)+m(w))_.

Using the Fourier decomposition of a and b, a#b can be written as a#b(z) = 1 (2πh)4n Z R2n dl Z R2n dmhe2hiσ(hDz,hDw)_ehi(l(z)+m(w))|_z=w_ˆ_a_h_(l)ˆ_b_h_(m) i = 1 (2πh)4n Z R2n dl Z R2n dmhe2hiσ(l,m)e i h(l+m)(z)aˆ_h(l)ˆb_h(m) i , so its Weyl quantisation becomes

(a#b)W(x, hD) = 1 (2πh)4n Z R2n dl Z R2n dmhe2hiσ(l,m)e i h(l+m)(x,hD)ˆa_h(l)ˆb_h(m) i = 1 (2πh)4n Z R2n dl Z R2n dmhehil(x,hD)e i hm(x,hD)ˆa_h(l)ˆb_h(m) i = aW(x, hD)bW(x, hD).

Symbols of the form a(x, ξ) = a(ξ) =P

αcαξα for certain constants cα have the property that

Opt(a)u(x) = 1 (2πh)n Z Rn dξ Z Rn dy " X α cαξαu(y)e i hhx−y,ξi # = 1 (2πh)n Z Rn dξ " X α cαξαuˆh(ξ)e i hhx,ξi # =X α cα(hD)αu(x).

This allows us to write equation (4.8) in integral form.

Lemma 4.17. (Integral form of a#b) Let a, b ∈S (R2n_{) and z ∈ R}2n_{, then}

a#b(z) = 1 (πh)2n Z R2n dw1 Z R2n dw2 h e−2ihσ(w1,w2)_{a(z + w} 1)b(z + w2) i (4.10)

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Proof. Let (w1, w2) = (y1, η1, y2, η2) ∈ R4n take the role of y, and let (z, w) = (x, ξ, y, η) ∈ R4n the role

of x, and (z0, w0) ∈ R4n take the role of ξ. Then a#b(z) = e2hiσ(hDz,hDw)_(a(z)b(w))| z=w = 1 (2πh)4n Z R4n d(w1, w2) Z R4n d(z0, w0)he2hiσ(z 0_,w0₎ ehih(z,w)−(w1,w2),(z0,w0)i_a(w 1)b(w2) i _z=w = Z R4n d(w1, w2)hFh−1 (z0, w0) 7→ e2hiσ(z 0_,w0₎ (z − w1, z − w2)a(w1)b(w2) i = Z R4n d(w1, w2)hFh−1 (z0, w0) 7→ e2hiσ(z 0_,w0₎ (−w1, −w2)a(z + w1)b(z + w2) i , where the inverse Fourier transform is given by

F−1 h (z, w) 7→ e2hi σ(z,w) (w1, w2) = 1 (2πh)4n Z R4n d(z, w)he2hi σ(z,w)e i hh(w1,w2),(z,w)i i = 1 (2πh)2n Z R2n d(x, η)he−2hihx,ηie i hhx,y1i_ehihη,η2i i · 1 (2πh)2n Z R2n

d(y, ξ)he2hihy,ξi_ehihy,y2i_ehihξ,η1ii

= 1 (2πh)2n Z Rn dη ehihη,η2i Z Rn dxhehih x 2,2y1i_e−hih x 2,ηi i · 1 (2πh)2n Z Rn dy ehihy,y2i Z Rn dξhehih ξ 2,2η1ie i hhy, ξ 2i i = 1 (2πh)2n Z Rn dηh2nFh x 7→ ehihx,2y1i_(η)ehihη,η2ii · 1 (2πh)n Z Rn dyh2nF_h−1ξ 7→ ehihξ,2η1i_(y)e−hihy,−y2ii = 1 (πh)ne 2i hhy2,η1i_· 1 (πh)ne −2i hhy2,η1i₌ 1 (πh)2ne 2i hσ(w1,w2)_.

Corollary 4.18. (# is associative) Let a, b, c ∈S (R2n_{), then (a#b)#c = a#(b#c).}

Proof. Using the integral form, we obtain: (a#b)#c(z) = 1 (2πh)4n Z R8n d ˜w1d ˜w2dw1dw2 h e−2ih(σ( ˜w1, ˜w2)+σ(w1,w2))_{a(z + ˜}_w 1+ w1)b(z + ˜w1+ w2)c(z + ˜w2) i a#(b#c)(z) = 1 (2πh)4n Z R8n d ˜v1d ˜v2dv1dv2 h e−2ih(σ( ˜v1, ˜v2)+σ(v1,v2))a(z + ˜v₁)b(z + ˜v₂+ v₁)c(z + ˜v₂+ v₂) i

It is easy to see that these integrals are equal by using the substitution v1 = ˜w1, v2 = ˜w2− w2, ˜v1 =

˜

w1+ w1, ˜v2= w2.

Definition 4.19. Let ϕ ∈S (Rn), N ∈ N, then we say ϕ = O_S(hN) if for all multi-indices α, β, there is a constant Cα,β > 0 such that

sup

x∈Rn

|xα∂βϕ(x)| ≤ Cα,βhN

as h → 0.

Theorem 4.20. Let a, b ∈S (R2n_{), and N ∈ N, then}

a#b(z) = N −1 X k=0 " 1 k! ih 2 k σ(Dz, Dw)k(a(z)b(w)) # z=w + O_S(hN) (4.11) as h → 0.

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Proof. First note that F−1_{(z, w) 7→ e}−2i hσ(z,w) (w1, w2) = h 4π 2n eih2σ(w1,w2)_,

the proof of which is similar to the calculation in the previous lemma. Now we have for all z ∈ R2n, a#b(z) = 1 (πh)2n Z R2n dw1 Z R2n dw2 h e−2ihσ(w1,w2)_{a(z + w} 1)b(z + w2) i = 1 (πh)2n Z R2n dw1 Z R2n dw2hF−1 (z, w) 7→ e−2ihσ(z,w) (w1, w2)eihz,w1+w2iˆa(w1)ˆb(w2) i = 1 (2π)4n Z R2n dw1 Z R2n dw2 h eih2σ(w1,w2)_eihz,w1+w2i_ˆ_a(w 1)ˆb(w2) i = 1 (2π)4n Z R2n dw1 Z R2n dw2 h ei(h2σ(w1,w2)+hz,w1+w2i)F (a ⊗ b)(w 1, w2) i . We will introduce the convenient notation Jz(h, a ⊗ b) := a#b(z) as well as P := ₂iσ(Dw0

1, Dw02). Then ∂hJz(h, a ⊗ b) = 1 (2π)4n Z R4n d(w1, w2) ei(h2σ(w1,w2)+hz,w1+w2i)i 2σ(w1, w2)F (a ⊗ b)(w1, w2) = 1 (2π)4n Z R4n d(w1, w2) h ei(h2σ(w1,w2)+hz,w1+w2i)F (P a ⊗ b)(w 1, w2) i = Jz(h, P a ⊗ b)

Consequently, ∂_hkJz(h, a ⊗ b) = Jz(h, Pka ⊗ b) for all k ∈ N. Taylor’s theorem around h = 0 now gives

for any N ∈ N that

a#b(z) = N −1 X k=0 hk k!Jz(0, P k_{a ⊗ b)} +h N N !Rz,N(h, a ⊗ b) where Rz,N(h, a ⊗ b) := N R1 0 dt[(1 − t) N −1_J

z(th, PNa ⊗ b)]. It is now left to show that Jz(0, Pka ⊗ b) is

indeed the required expression and that the rest term is indeed O_S(hN_{), i.e. |R}

z,N(h, a ⊗ b)| is bounded independent of h. • Jz(0, Pka ⊗ b) = 1 (2π)4n Z R4n d(w1, w2) " F (w0₁, w0₂) 7→ i 2 k σ(Dw0 1, Dw20) k_a(w0 1)b(w 0 2) ! (w1, w2)eihz,w1+w2i # = i 2 k σ(Dz, Dw)ka(z)b(w) _z=w • |Rz,N(h, a ⊗ b)| = N Z 1 0 dt(1 − t)N1_J z(th, pNa ⊗ b) ≤ CNkF (PNa ⊗ b)kL1 ≤ CN max |α|≤n+1k∂ α_PN_{a ⊗ bk} L1 ≤ CN max |α|≤N +n+1k∂ α_{a ⊗ bk} L1, by lemma3.6.

Corollary 4.21. Let a, b ∈S (R2n_{), then}

a#b = ab + h 2i{a, b} + OS(h 2₎ _(4.12) and [aW(x, hD), bW(x, hD)] = h i{a, b} W_{(x, hD) + O} S(h3) (4.13)

where [A, B] := AB − BA is the commutator and {f, g} :=Pn

j=1(fξjgxj− fxjgξj) is the Poisson bracket

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Proof. a#b(z) = a(z)b(z) + ih 2σ(Dz, Dw)a(z)b(w) _z=w + O_S(h2) = a(z)b(z) + h 2i(∂y∂ξ− ∂x∂η)a(x, ξ)b(y, η) _{(x,ξ)=(y,η)} + O_S(h2) = a(z)b(z) + h 2i{a, b}(z) + OS(h 2₎ [aW(x, hD), bW(x, hD)] = (a#b)W(x, hD) − (b#a)W(x, hD) = h i{a, b} W_{(x, hD) −} h2 8 σ(Dz, Dw) 2_{(a(z)b(w) − b(z)a(w))} _z=w + O_S(h3) = h i{a, b} W_{(x, hD) + O} S(h3)

4.3 Symbol classes

In this subsection we will prove that aW(x, hD) : L2 → L2 _{is well-defined as a bounded operator for}

certain symbol classes that are larger thanS (R2n).

Definition 4.22. (Order functions) A measurable function m : R2n_{−→ (0, ∞) is called an order function}

if ∃C, N ∈ R such that ∀w, z ∈ R2n_,

m(w) ≤ Chz − wiNm(z), where hzi :=p1 + |z|2_.

Proposition 4.23. Let m, m1, and m2 be order functions and let a ∈ [0, ∞), then ma, 1/m, m1+ m2,

and m1m2 are order functions as well. Moreover, mk,l defined by

mk,l(z) := hxik+ hξil (4.14)

is an order function for all k, l ∈ R.

Proof. Since m is an order function, there are constants C, N ∈ R such that m(w) ≤ Chz − wiN_{m(z) for}

all w, z ∈ R2n_{. Then} ma(w) ≤ Cahz − wiN a_ma_{(z), and} 1 m(z) ≤ Chz − wi N 1 m(w).

Now let Cj, Nj ∈ R such that mj(w) ≤ Cjhz − wiNjmj(z) for j = 1, 2 and all w, z ∈ R2n, and assume

without loss of generality that N1≤ N2. Then

(m1+ m2)(w) = m1(w) + m2(w) ≤ C1hz − wiN1m1(z) + C2hz − wiN2m2(z) = hz − wiN2_C 1hz − wi−(N2−N1)m1(z) + C2m2(z) ≤ hz − wiN2_(C 1m1(z) + C2m2(z)) ≤ max(C1, C2)hz − wiN2(m1+ m2)(z), (m1m2)(w) = m1(w)m2(w) ≤ C1C2hz − wiN1+N2(m1m2)(z).

Now it is only left to show that m(w) := hxi is an order function. Note that m(w) = hyi =p1 + |y|2_≤p_{1 + (|y − x| + |x|)}2

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We will consider two cases: (a) |x||y − x| ≤ 1, and (b) |x||y − x| ≥ 1. Then m(w)

(a)

≤ p1 + |x − y|2_{+ |x|}2_{+ 2 ≤}√₂p

1 + |x − y|2_{+ 1 + |x|}2

≤√2 (hy − xi + hxi) ≤ 2√2 max(hy − xi, hxi) ≤ 2√2hy − xihxi = 2√2hy − xim(z),

m(w)

(b)

≤p1 + |y − x|2_{+ |x|}2_{+ 2|y − x|}2_|x|2_≤√₂p_{(1 + |y − x|}2_{)(1 + |x|}2₎

=√2hy − xihxi =√2hy − xim(z).

Definition 4.24. (Symbol classes) Let m : R2n_{→ (0, ∞) be an order function and let δ ≥ 0, then}

Sδ(m) := {a ∈ C∞| ∀α, ∃Cα> 0; |∂αa| ≤ Cαh−δ|α|m}. (4.15)

We shall write S(m) := S0(m), Sδ := Sδ(1), and S := S0(1). Note that

sup

z,h

m < ∞ =⇒ Sδ(m) ⊆ Sδ,

inf

z,hm > 0 =⇒ Sδ ⊆ Sδ(m).

The constant δ ≥ 0 is relevant in case we want to study aW_{(x, hD) in the limit h → 0. Of course, the}

quantisation formula (4.1) itself already depends on h. By rescaling ˜x := h−12x, ˜ξ := h− 1

2ξ, ˜y := h− 1 2y,

˜

u(˜x) := u(x) = u(h12x), ˜˜ a(˜x, ˜ξ) := a(x, ξ) = a(h12x, h˜ 12ξ), we obtain˜

aW(x, hD)u(x) = 1 (2πh)n Z Rn dξ Z Rn dy ehihx−y,ξia x + y 2 , ξ u(y) = 1 (2πh)nh nZ Rn d ˜ξ Z Rn d˜y eih˜x−˜y, ˜ξia h12x + ˜˜ y 2 , h 1 2ξ u(h12y)˜ = 1 (2π)n Z Rn d ˜ξ Z Rn d˜y eih˜x−˜y, ˜ξia˜ ˜x + ˜y 2 , ˜ξ ˜ u(˜y) = ˜aW(˜x, D)˜u(˜x).

Let δ ≥ 0, let m be an order function, and let a ∈ Sδ(m). Then for all multi-indices α, the rescaled

function ˜a satisfies |∂αa| = h˜ 12|α||∂αa| ≤ C_αh|α|( 1

2−δ)m. This is unbounded as h → 0 for δ > 1

2, so from

now on we will always assume that 0 ≤ δ ≤ 1₂.

Proposition 4.25. Let δ ≥ 0 and t ∈ [0, 1]. Let m be an order function, and let a ∈ Sδ(m). Then

Opt(a) :S (Rn) →S (Rn)

is a continuous linear operator.

Proof. Let u ∈ S (Rn). We want to prove that x 7→ Opt(a)u(x) is again a Schwartz function. We

will first prove that supx∈Rn|Opt(a)u(x)| < ∞. Then, for 1 ≤ j ≤ n, we will apply this to the cases

xjOpt(a)u(x) and ∂jOpt(a)u(x) by writing these as the finite sum of functions of the form Opt(b)v(x)

for certain b ∈ Sδ(m), v ∈S (Rn).

Let C, N > 0 such that m(w) ≤ Chz − wiN_{m(z) for all w, z ∈ R}2n_{. Then for all 1 ≤ j ≤ n,}

hDyje i hhx−y,ξi= −ξ_je i hhx−y,ξi, and hDξje i hhx−y,ξi_{= (x}_j− y_j_)ehihx−y,ξi_.

So for the operators L1and L2defined by

L1:= 1 − hξ, hDyi 1 + |ξ|2 = 1 − hξ, hDyi hξi2 , L2:= 1 + hx − y, hDξi 1 + |x − y|2 = 1 + hx − y, hDξi hx − yi2 ,

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the identities L1e i hhx−y,ξi= L₂e i hhx−y,ξi= e i

hhx−y,ξihold. Then we obtain

Opt(a)u(x) =

1 (2πh)n

Z

R2n

d(ξ, y)hehihx−y,ξia (tx + (1 − t)y, ξ) u(y)

i = 1 (2πh)n Z R2n d(ξ, y)hLN +n+1₁ ehihx−y,ξi a (tx + (1 − t)y, ξ) u(y)i = 1 (2πh)n Z R2n d(ξ, y) " ehihx−y,ξi N +n+1 X k=0 hξ, hDyik